Elastic Surface Model For Beta‐Barrels: Geometric ... Surface Model For Beta-… · minimizers of...
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R E S E A R CH AR T I C L E
Elastic Surface Model For Beta-Barrels: Geometric,Computational, And Statistical Analysis
Magdalena Toda | Fangyuan Zhang | Bhagya Athukorallage
Department of Mathematics and Statistics,
Texas Tech University, Lubbock, Texas
79409
Correspondence
Bhagya Athukorallage, Department of
Mathematics and Statistics, Texas Tech
University, Lubbock, TX 79409.
Email: [email protected]
AbstractOver the past 2 decades, many different geometric models were created for beta barrels, including,
but not limited to: cylinders, 1-sheeted hyperboloids, twisted hyperboloids, catenoids, and so forth.
We are proponents of an elastic surface model for beta-barrels, which includes the minimal surface
model as a particular case, but is a lot more comprehensive. Beta barrel models are obtained as
numerical solutions of a boundary value problem, using the COMSOL Multiphysics Modeling Soft-
ware. We have compared them against the best fitting statistical models, with positive results. The
geometry of each individual beta barrel, as a rotational elastic surface, is determined by the ratio
between the exterior diameter and the height. Through our COMSOL computational modeling, we
created a rather large variety of generalized Willmore surfaces that may represent models for beta
barrels. The catenoid is just a particular solution among all these shapes.
K E YWORD S
beta barrel, generalized Willmore energy, mean curvature, minimal surface, protein structure
1 | INTRODUCTION
In biochemistry, biophysics and mathematical biology, secondary struc-
tures represent the main types of local surfaces (shapes) corresponding
to biopolymers (eg, proteins and nucleic acids). On a finer level, the
atomic positions in 3D space are said to form the tertiary structure.
The most common secondary structures are the alpha helices. The
second most-common are the beta sheets and beta barrels. A beta bar-
rel is a collection of beta-sheets that twist and coil in a shape that can
be described as a smooth surface of revolution which resembles a bar-
rel. In this structure, the first strand is hydrogen bonded to the last.
Beta-strands in beta-barrels are typically arranged in an antiparallel
fashion. Barrel structures are commonly found in proteins that span
cell membranes and in proteins that bind hydrophobic ligands in the
barrel center.
A beta barrel represents an ideal smooth surface described by the
beta strands. The most common type, the so called up-and-down bar-
rel, is considered to be a surface of revolution that is topologically
equivalent to a cylinder. Up-and-down barrels consist of a series of
beta strands, each of which is hydrogen-bonded to the strands immedi-
ately before and after it in the primary sequence. Beta-strands in
beta-barrels are typically arranged in an antiparallel fashion, but some
proteins, such as the green fluorescent protein (GFP), are characterized
by beta barrels formed with both parallel and antiparallel beta strands.
Beta barrel structures (named for their resemblance to the barrels that
hold liquids) are commonly found in porins and other proteins that
span cell membranes, and in proteins that bind hydrophobic ligands in
the barrel center, as in lipocalins.
In many cases, the strands contain alternating polar and hydropho-
bic amino acids, so that the hydrophobic residues are oriented toward
the interior of the barrel to form a hydrophobic core and the polar resi-
dues are oriented toward the outside of the barrel on the solvent-
exposed surface. Porins and other membrane proteins containing beta
barrels reverse this pattern, with hydrophobic residues oriented toward
the exterior where they contact the surrounding lipids, and hydrophilic
residues oriented toward the interior pore.
The polyhedral, discrete skeleton of beta-barrels can be classified
in terms of 2 integer parameters: the number of strands in the beta-
sheet, n, and the “shear number”, S, a measure of the stagger of the
strands in the beta-sheet. These 2 parameters (n and S) are related to
the inclination angle of the beta strands relative to the axis of the bar-
rel. To us, the number of strands and the sheer number are irrelevant.
We consider the best fitting smooth surface for this polyhedral molecu-
lar skeleton, to be what we refer to as beta barrel.
Several models were proposed for beta sheets and beta barrels.
Among them, we recall, in chronological order: the twisted 1-sheeted
Proteins. 2018;86:35–42. wileyonlinelibrary.com/journal/prot VC 2017Wiley Periodicals, Inc. | 35
Received: 20 July 2017 | Accepted: 11 October 2017
DOI: 10.1002/prot.25400
hyperboloid (see,1 1984), followed by the usual 1-sheeted hyperboloid
(see,2 1988), and much later, the catenoid as a best-fit (see,3 2005).
Over time, all the above-mentioned surfaces have been tried as “best
models” for beta barrels. We became aware of the fact that none of
these models is satisfactory. A few authors presented arguments based
on experimental data that the beta sheet structures in proteins are the
result of the tendency to minimize surface areas, and should therefore
be very close to minimal surfaces. For a large diversity of aminoacids,
the mean curvature was experimentally measured and it turned out to
be close to a specific constant, which is small in absolute value, but not
negligible. A decade ago, Koh and Kim3 have proposed the model that
all beta-sheet structures “are almost minimal surfaces”. By this,
they meant that their mean curvatures are very small, but the term
almost-minimal has no mathematical foundation.
We would like to make it clear, however, that while the mean cur-
vature could be small in absolute value for some beta barrels, it may be
far away from zero for others, and it may also vary significantly from a
point to another of the same barrel. Koh and Kim stated: “The fact that
the commonly used models for some beta-sheet surfaces (ie, the
hyperboloid and strophoid) have very small mean curvatures (under
0.05) supports our model”. For example, for the following enzymes:
glycolate-oxidase, taka-amylase, and aldolase, the mean curvature H,
measured experimentally for beta-sheets, is approximately H50.039
(for each of them). In reference,4 the authors presented the mean cur-
vature values for several aminoacids, and are depicted in Table 1. On
the other hand, the mean curvature values may change significantly
from a type of protein to another.
Moving in a new direction, that can give us a better view, we are
proponents of an “elastic surface model for beta-barrels”, which includes
the minimal surface model as a particular case, but is a lot more compre-
hensive. Such a model has already been proposed in part by5 for beta
sheets with antiparallel strands. We study beta barrels as elastic surfa-
ces, and obtain them as numerical solutions of a boundary value prob-
lem—while comparing them against the best fitting statistical models. It
is important to remark that catenoids represent the only minimal surfa-
ces of revolution, and in particular, they are Willmore-type surfaces
(elastic surfaces) [14]. As such, they are included in our model. Through
our COMSOL computational modeling, we created a rather large variety
ofWillmore-type surfaces that may represent models for beta barrels.
As a relevant real-world application of high relevance and actuality,
observe the model of the beta barrel of GFP in Figure 1 and remark its
unduloidal shape (Delaunay-type unduloid). Martin Chalfie, Osamu Shimo-
mura, and Roger Y. Tsien were awarded the 2008 Nobel Prize in Chemis-
try for their discovery and development of the GFP model. On the other
hand, Helfrich [10] has introduced the curvature energy per unit area, cor-
responding to bio-membranes (lipid bilayers) as:
Elb5ð
M
kcð2H1c0Þ21�kK dS; (1)
where kc and �k represent specific rigidity constants, H and K are the mean
and Gaussian curvatures of the surfaceM, respectively, while c0 is the so-
called spontaneous curvature.
We would like to mention that this type of generalized bending
energy (at times referred to as Helfrich-type energy) has been
considered for other types of elastic membranes in biophysics, as well,
e.g. [6], [7], [11], [12]. Following the model proposed by Helfrich, two
other scientists, S. Choe and X.S. Sun, proposed a similar elastic model
for anti-parallel beta sheets, which was published in 2007 in the Bio-
physical Journal5—based on a bending energy, namely:
Elb5ð
M
½k ðH1c0Þ21�kK� dS; (2)
where dS is the infinitesimal area, c0 is the “preferred curvature of the
surface” as they call it, and k and ~k are bending moduli that “relate the
energy change with changes in mean and Gaussian curvatures”.
Our elastic surface model for beta barrels is similar to this model,
with the exception of an added constant term that comes from the
superficial tension combined with a stress tensor, and with the
additional assumption that c0 is negligible. We therefore write
Eb5ð
M
½kH21~kK1l� dS; (3)
TABLE 1 Mean curvature values of beta barrels for several typesof proteins (from reference4)
Protein
Averageof meancurvature
SD of meancurvature
Triose phosphate isomerase 0.040 0.021
Taka-amylase 0.035 0.007
Glycolate-oxidase 0.035 0.007
Trimethanolaminedehydrogenase
0.037 0.013
Cytochrome b2 0.033 0.005
Aldolase 0.035 0.112
FIGURE 1 Beta Barrel of the GFP (from jelly fish) by scientistsMartin Chalfie, Osamu Shimomura and Roger Tsien—Nobel prizerecipients (courtesy of the American Association of ClinicalChemistry) [Color figure can be viewed at wileyonlinelibrary.com]
36 | TODA ET AL.
and call this type of energy generalized Willmore energy (GW energy),
or generalized bending energy. The Euler-Lagrange equation corre-
sponding to the GW energy functional can be written as the following
(GWE):
DgðHÞ12HðH22K2�Þ50; (4)
where �5l=k and Dg represents the Laplace-Beltrami operator corre-
sponding to the metric g that is naturally induced by the surface param-
eterization. We are interested in rotational surfaces that represent
minimizers of GW energy (that is, solutions to its corresponding Euler-
Lagrange equation described above). These represent our general mod-
els for beta barrels. Hereby, we are recalling the basics of a computa-
tional study that we have performed on this type of surfaces, in.7
Consider a Cartesian system of axes of coordinates x, y, z in R3 and the
circles C1, C2 of the same radius a, centered at ð21;0;0Þ and (1, 0, 0),
situated in planes orthogonal to the x axis. Consider all regular surfaces
of revolution of annular-type with boundary C1 [ C2. Assume that
among all these surfaces, there exists at least a surface M minimizing
the GWE. This surface in assumed embedded in R3 and admitting the
representation
M : 5fx; uðxÞcosu; uðxÞsinug : x 2 ½21;1�; u 2 ½0;2p�;
where u 2 C4ð½21;1�; ð0;1ÞÞ represents the profile function. Then,
the surface M is a solution of the following boundary value problem:
DH12HðH22K2�Þ50 onM; where �5lk
(5)
H50 on oM5C1 [ C2; (6)
uð61Þ5a: (7)
In these assumptions, there exists a positive value a� (a� � 1:5089)
that is independent from the value of �, such that
a If 0<a<a�, then GWE admits NO minimal solution, that is, any
solution satisfies: H50 on oM and H 6¼ 0 on M n ðoMÞ.
FIGURE 2 Solutions to our boundary value problem, as H(x), and corresponding profile curves u(x) for different a values and for fixed� value [Color figure can be viewed at wileyonlinelibrary.com]
TODA ET AL. | 37
b If a5a�, then GWE admits exactly 1 minimal solution (a unique
catenoid that exclusively depends on a�).
c If a>a�, then GWE admits exactly 2 minimal solutions (2 catenoids
whose equations exclusively depend on a).
The proof is straight-forward, as based on elementary arguments, and
can be found in.7 Such a Dirichlet boundary value problem was posed
for the Willmore equation in [9].
Further, we have analyzed the a-family of solutions that corre-
sponds to various fixed values of �. We were able to construct corre-
sponding families of solutions using COMSOL Multiplysics, see fig. 2
and 3. On the other hand, of course, each solution to our boundary
value problem (and in particular each catenoidal solution) can actually
be represented in COMSOL, if we choose the unique value a appropri-
ately, and deal with the solution branching (in order to graph all corre-
sponding solutions u if that is the case).
Remark the shapes obtained for the profile u(x) as solutions to the
boundary value problem associated to GWE, in all the figures pre-
sented in this article: they resemble either a catenary, or an undulary—
thus generating catenoidal and unduloidal GW surfaces of revolution.
Due to the physical nature of our boundary value problem, the nodoi-
dal solutions are absent, but nodoidal solutions would certainly be pres-
ent for other types of boundary value conditions of the GWE.
Following our analysis, for each and every value of a that is higher than
a�, there exist 3 distinct solutions, namely 2 catenoidal profiles and a
non-minimal solution—which could be unstable (that is, not a local mini-
mizer of the energy). Remark that catenoids represent global minimiz-
ers, as Deckelnick and Grunau8 showed in a recent article. For the
classical Willmore case �50, authors proved that the non-minimal
solution is contained between the 2 catenoids, and it is unstable. Our
numerical analysis on the stability of the solutions to the GWE is in
progress (Figures 2 and 3).
FIGURE 3 Solutions to our boundary value problem, as H(x), and corresponding profile curves u(x) for different a values and for fixed � value[Color figure can be viewed at wileyonlinelibrary.com]
38 | TODA ET AL.
Beta barrels are hereby represented as solutions of the boundary
value problem of the GW equation, GWE. Let D represent the “exterior
diameter” of the beta barrel, which is defined as the diameter of the
end-circles (what we call the imaginary top and bottom of the beta bar-
rel). Let h represent the distance between these two circles, which is
informally referred as the height of the beta barrel. Considering the
mathematical model of the beta barrel (solution of our boundary value
problems), its corresponding height will be 2 (distance between 21 and
1 on the x axis). Correspondingly, the diameter of the “bottom and top
circles” will be 2a. Therefore, the meaning of the a parameter from the
mathematical model is that of the ratio between diameter and height,
namely D/h, of the real-life beta barrel.
Elasticity theory provides enough reasons for the validity of our
computational PDE model. On the other hand, we decided to
strengthen our arguments by making an unbiased comparison between
our computational model (solution of the boundary value problem) and
the statistical models that correspond to protein databases.
2 | MATERIALS AND METHODS
2.1 | Beta barrel structure dataset from GFP class
To test the performance of the elastic surface model, we used the GFP
Ca coordinates dataset from Protein Data Bank. Out of the 54 proteins,
we excluded 10 with more than 1 domain, and extracted the structure
information using Dictionary of Protein Secondary Structure files. We
considered only “E” segments that form intradomain b-ladders as
b-strands.
2.2 | Least square estimation using 3 models
We used least square estimation (LSE) to compare the elastic surface
model with 2 other models: catenoid and cylinder. In Cartesian system,
the coordinates of a catenoid (x, y, z) need to satisfy the following
equations:
r5a coshza
� �
x5r cos ð2pfÞy5 rsin ð2pfÞ
8>>><>>>:
9>>>=>>>;
We used 211 equally spaced a values from 0.4 to 2.5. For each a
value, we generated 300 equally spaced points from 21 to11 for the
variable z, and 100 points from 0 to 1 for f. Therefore, we obtained
30,000 points for each catenoid with a different a value. The diameter/
height ratio a cosh 1a
� �ranges from 1.5 to 2.7. In Cartesian system, the
coordinates of a cylinder (x, y, z) need to satisfy the following
equations:
r is a constant for given z
x5r cos ð2fpÞy5r sin ð2fpÞ
8>><>>:
9>>=>>;
For the variable r, we used 100 equally spaced points, namely
from from 0.7 to 1.7. For each r value, we generated 300 equally
spaced points from 21 to11 for z and, respectively, 100 points
from 0 to 1 for f. Therefore, we get 30, 000 points for each cylinder
with a different r value. The diameter/height ratio r ranges from 0.7
to 1.7. For the elastic surface model, we used 89 equally spaced �
values from 0 to 0.88, and 48 equally spaced diameter/height values
a from 0.7 to 1.17. For each of � and a combinations, we generated
300 equally spaced points from 21 to11 for z, and 100 points from
0 to 1 for f, respectively. Therefore, by using COMSOL Multiplysics,
we obtained 30,000 points for each elastic surface, with different �
and a values.
The procedure to get the LSE for the 3 models can be described as
follows:
1. S denotes one simulated dataset. D denotes the Ca coordinates
data for a certain protein. Denote n as the number of points in
dataset D.
2. Randomly select n points from S, denoted as s. Denote d5D.
3. Run Procrustes analysis for optimal modeling, up to translating,
rotating, and scaling d to superimpose to s. d is updated as the
transformed data. The scale value is recorded.
4. Search in dataset S to find n points closest to each of the points in
d, denoted as s.
5. Repeat steps steps 3) and 4) 10,000 times, and record d with
the least sum of squares for error (SSE) from the subset s in
the iteration, denoted as d0 and s0 with series of scale values
as c1; c2; . . . ; ct.
6. Denote d15d0=ðc1c2 � � � ctÞ as the transformed real data in the
original scale. Apply the same translating, rotating, and scaling
procedure on S as the one to optimally superimpose s0 to d1.
The result is denoted as S1.
7. Apply the Iterative Closest Point method to match S1 to d1, result-
ing in rotated data S2. Then get correspondences in S2 using near-
est neighbor search, denoted as s2.
8. Calculate SSE between s2 and d1.
For each model, we run the above procedure to find the best parameter
value to achieve the smallest SSE. The results are shown in the following
table.
3 | RESULTS AND DISCUSSION
3.1 | Statistical model comparison with the
computational models in COMSOL
We can see from Table 2 that almost all of the roots of mean squares
for error (RMSE) are smaller than 3, by fitting the 3 models, except pro-
teins 1f09 and 1qyf. It turns out that the 3D structures of these 2 pro-
teins are quite different from the others. Further investigation is
needed in order to fit the two proteins using other more proper mod-
els. For all the other proteins, the elastic surface model achieves its
smallest RMSE when the estimated diameter/height ratio ranges
between 0.83 and 0.89. The cylindrical beta barrel model corresponds
TODA ET AL. | 39
TABLE 2 LSE results for GFP by using 3 models
Catenoid Cylinder Elastic surface
a diam./ht. RMSE diam./ht. RMSE epsilon diam./ht. RMSE
1bfp 0.76 1.518 2.582 0.73 1.371 0.67 0.85 1.318
1c4f 0.8 1.511 2.569 0.74 1.354 0.68 0.86 1.306
1cv7 0.81 1.510 2.496 0.77 1.373 0.69 0.88 1.323
1ema 0.77 1.516 2.539 0.71 1.356 0.64 0.85 1.296
1emb 0.76 1.518 2.578 0.73 1.375 0.67 0.86 1.318
1emf 0.79 1.512 2.527 0.72 1.359 0.63 0.85 1.293
1emg 0.79 1.512 2.549 0.72 1.351 0.66 0.85 1.300
1emk 0.76 1.518 2.549 0.7 1.339 0.64 0.84 1.295
1eml 0.8 1.511 2.560 0.72 1.373 0.62 0.85 1.327
1emm 0.76 1.518 2.540 0.72 1.380 0.66 0.87 1.322
1f0b 0.76 1.518 2.575 0.72 1.369 0.71 0.88 1.316
1f09 0.4 2.453 5.128 0.56 4.564 0.02 0.7 4.566
1huy 0.75 1.521 2.556 0.74 1.411 0.63 0.88 1.345
1jby 0.77 1.516 2.590 0.74 1.361 0.67 0.87 1.321
1jbz 0.78 1.514 2.506 0.74 1.362 0.63 0.89 1.277
1kyp 0.77 1.516 2.566 0.71 1.404 0.66 0.87 1.348
1kyr 0.76 1.518 2.558 0.71 1.394 0.65 0.86 1.334
1kys 0.8 1.511 2.560 0.76 1.376 0.7 0.85 1.323
1mem 0.75 1.521 2.532 0.75 1.361 0.65 0.86 1.292
1myw 0.69 1.551 2.582 0.72 1.402 0.64 0.86 1.349
1oxd 0.74 1.525 2.598 0.72 1.387 0.68 0.86 1.337
1oxe 0.74 1.525 2.565 0.71 1.377 0.7 0.87 1.326
1oxf 0.75 1.521 2.587 0.74 1.401 0.7 0.86 1.350
1q4a 0.78 1.514 2.546 0.74 1.370 0.66 0.87 1.316
1q4b 0.75 1.521 2.571 0.72 1.372 0.64 0.88 1.322
1q4c 0.75 1.521 2.553 0.73 1.377 0.66 0.88 1.322
1q4d 0.78 1.514 2.568 0.72 1.371 0.63 0.85 1.321
1q4e 0.76 1.518 2.549 0.74 1.380 0.66 0.86 1.332
1q73 0.72 1.534 2.554 0.73 1.371 0.65 0.86 1.324
1qxt 0.78 1.514 2.524 0.74 1.344 0.67 0.86 1.281
1qy3 0.73 1.529 2.569 0.74 1.369 0.68 0.86 1.309
1qyf 0.4 2.453 5.108 0.51 4.022 0 0.7 4.124
1qyo 0.79 1.512 2.554 0.76 1.419 0.68 0.86 1.350
1qyq 0.78 1.514 2.526 0.7 1.412 0.66 0.87 1.342
1rm9 0.78 1.514 2.513 0.74 1.406 0.63 0.88 1.323
1rmm 0.77 1.516 2.536 0.72 1.387 0.67 0.87 1.318
1rmo 0.74 1.525 2.543 0.73 1.388 0.65 0.86 1.326
1rmp 0.74 1.525 2.538 0.74 1.372 0.63 0.88 1.298
(Continues)
40 | TODA ET AL.
to a slightly higher RMSE, and similar diameter/height ratios. The cate-
noidal beta barrel model corresponds to a much higher RMSE and very
different estimates of diameter/height ratios.
The “practical reason” for the much better fit of the elastic
unduloidal-type model (when compared with its catenoidal counter-
part) is the following: On one hand the GFP structure has a small
diameter/height ratio, and on the other hand, while both cylinder
model and elastic model can fit structures with various diameter/height
ratios, the catenoidal model can only allow that ratio to be greater than
(or equal to) the value a�51:5089.
The cylindrical model is close enough to the unduloidal-type elastic
surface, and definitely more appropriate/accurate than the catenoid.
We should not forget that, for the case when c0 is not negligible, the
GWE equation has H5 c0 as a stable solution. Therefore, the constant
mean curvature (CMC non-zero) model may include circular cylinders,
as well as true CMC unduloids. Our elastic model, together with the
statistical analysis, now make it completely clear why the GFP protein
model obtained in 2008 by Chalfie, Shimomura and Tsien possesses
neither a catenoidal shape, nor a round cylindrical one, but an unduloi-
dal one (see Figure 4). The shape of the profile function u of the beta
barrel is always determined by the ratio between the end-circle diame-
ter and height (which is a). To complete the discussion, we also studied
and modeled a protein example 1cka with larger diameter/height ratio.
The results in Table 3 show that because the diameter/height ratio is
larger for protein 1cka, catenoid model can fit better than cylinder and
elastic surface model.
4 | CONCLUSION
Our elastic surface model for beta barrels proves itself to be more
appropriate than the individual models that were historically used. Phys-
ical arguments led us to beta barrel models as solutions to a boundary
value problem associated to a general Willmore equation. In addition,
our elastic surface model satisfies the requirements of the statistical
analysis, as a much better fit than previous models. This is a fortunate
case, in which numerical solutions obtained via COMSOL Multiphysics
(based on finite element methods) have been tested via statistical analy-
sis, and the results were highly in favor of the elastic model.
ORCID
Bhagya Athukorallage http://orcid.org/0000-0002-1240-223X
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Catenoid Cylinder Elastic surface
a diam./ht. RMSE diam./ht. RMSE epsilon diam./ht. RMSE
1rrx 0.76 1.518 2.573 0.72 1.385 0.66 0.86 1.334
1s6z 0.79 1.512 2.598 0.71 1.418 0.67 0.86 1.365
2emd 0.76 1.518 2.537 0.74 1.402 0.6 0.87 1.321
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1cka 1.15 1.613 0.877 1.22 1.096 0.65 1.16 1.045
FIGURE 4 LSE for GFP 1myw. Black squares represent real datapoints. Red, green, and blue dots represent the correspondingfitted elastic surface model, catenoid model, and cylinder model
TODA ET AL. | 41
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How to cite this article: Toda M, Zhang F, Athukorallage B.
Elastic Surface Model For Beta-Barrels: Geometric, Computa-
tional, And Statistical Analysis. Proteins. 2018;86:35–42.
https://doi.org/10.1002/prot.25400
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